
where \(c\)  is the speed of light. Note that \(\alpha _T\)  therefore is a measure of the `tensor speed excess', i.e. it is positive for \(c_{\rm GW}> c\)  and negative when \(c_{\rm GW}< c\) . We will remain agnostic here and will not assume anything about the sign of \(\alpha _T\)  in what follows.It is useful to recall that for a spontaneously Lorentz-breaking solution, \(c_{\rm GW}\le c\)  is not imposed by causality requirements – in fact the opposite can be the case, see e.g. [1]}, [2]} and references therein.
Constraints from the cosmic microwave background and large scale structure (see [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}, [16]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]}
and references therein), corresponding to frequencies \(f \sim 10^{-18} - 10^{-14} \; {\rm Hz}\) , require \(|\alpha _T| \lesssim {\cal O}(1)\) . Binary pulsars, in particular the Hulse-Taylor binary, impose \(|\alpha _T| \lesssim {\cal O}(10^{-2})\)  for frequencies \(f \sim 10^{-4} \; {\rm Hz}\)  [30]}. Gravitational wave observations of the afore-mentioned GW170817 event then place an extremely tight bound of \(|\alpha _T| \lesssim {\cal O}(10^{-15})\)  for frequencies \(f \sim 10^1 - 10^{3} \; {\rm Hz}\)  [31]}, [32]}, [33]}, [34]}, [35]} , while the presence of high energy cosmic rays (and inferred absence of gravitational Cherenkov radiation) leads to a \(\alpha _T\gtrsim -{\cal O}(10^{-15})\)  bound at energies of \(\sim 10^{10}\)  GeV [36]}, i.e. for very high frequencies \(f \sim 10^{34} \; {\rm Hz}\) . Note that the last bound does depend on the sign of \(\alpha _T\) . Finally, several forecasted bounds exist for future LISA observations [37]}, [38]}. Since these are particularly relevant in the context of this paper, we will discuss them in more detail in section .
The frequency-dependence of the bounds on \(\alpha _T\)  is particularly important, since the theoretical predictions for this parameter are also frequency-dependent. More specifically, for large classes of dark-energy related models that yield a significantly non-zero \(\alpha _T\)  on cosmological scales, i.e. at very low frequencies, one naturally expects a transition back towards \(\alpha _T= 0\)  close to or somewhat below \(10^2 \; {\rm Hz}\)  [39]}. This means that the tight GW170817 constraint on \(\alpha _T\)  is perfectly consistent with models that yield a detectable, non-zero \(\alpha _T\)  at lower frequencies.
